The document is a presentation on topology given by Nate Black at Bob Jones University on 11/19/07. It discusses the history of topology including contributors like Euler and Möbius. It covers topological concepts like the Königsberg bridge problem, the Möbius strip, and the Klein bottle. It also defines key topological concepts such as topological spaces, open and closed sets, connectedness, compactness, and applications to fields like network theory and knot theory.

1. Topology Senior Math Presentation Nate Black Bob Jones University 11/19/07

3. Königsberg Bridge Problem J. J. O’Connor, A history of Topology

4. Königsberg Bridge Problem A B C D A graph has a path traversing each edge exactly once if exactly two vertices have odd degree . 3 D 3 C 5 B 3 A Degree Vertex

5. Königsberg Bridge Problem A B C D A graph has a path traversing each edge exactly once if exactly two vertices have odd degree . 2 D 3 C 4 B 3 A Degree Vertex

8. Möbius Strip Plus Magazine ~ Imaging Maths – Inside the Klein Bottle

11. Klein Bottle Plus Magazine ~ Imaging Maths – Inside the Klein Bottle

12. General Topology Overview

19. Connectedness

31. Path Connected 0 1

34. Simple Connectedness (0,1) (1,1) (1,0) time space Mile marker 0 Mile marker 1 Time 0: Mile marker 0 Mile marker 1 Time 1:

35. Simple Connectedness (0,1) (1,1) (1,0) time space

39. Simply Connected Exit Entrance

40. Simply Connected

41. Compactness

45. Compactness 0 5 -5 2 is no longer covered Remove any interval

49. Applications

52. Applications Matt Black

53. Applications 2006 Encyclopædia Britannica

61. Applications + work – work 0 total work

62. Questions